Monodromy representations of $p$-adic differential equations in families

TL;DR

This paper extends the local monodromy theorem to families of $p$-adic differential equations over nonarchimedean fields, with applications in $p$-adic cohomology and Hodge theory.

Contribution

It introduces a relative version of the local monodromy theorem and applies it to simplify proofs and extend results in $p$-adic cohomology and Hodge theory.

Findings

A relative local monodromy theorem for $p$-adic differential equations.

Simplified proof of semistable reduction for overconvergent $F$-isocrystals.

Multivariate local monodromy theorem in the style of Drinfeld's lemma.

Abstract

We derive a relative version of the local monodromy theorem for ordinary differential equations on an annulus over a mixed-characteristic nonarchimedean field, and give several applications in $p$-adic cohomology and $p$-adic Hodge theory. These include a simplified proof of the semistable reduction theorem for overconvergent $F$-isocrystals, a relative version of Berger's theorem that de Rham representations are potentially semistable, and a multivariate version of the local monodromy theorem in the style of Drinfeld's lemma on fundamental groups.